Let's call the supplementary angle y.
Then, as you noticed, x + y = 180.
Since the areas are proportional to the angles, we can just reword the rest in terms of angles. i.e.
"The sum of the angles of the unshaded regions is equal to the average of the angles of the shaded regions"
The sum of the unshaded angles is x + x = 2x.
The average of the shaded angles is (y + y)/2 = y.
These are equal, so 2x = y.
Combining this with x + y = 180 will lead you to your desired result.